How did he get the nickname? It wasn't for his prolific (for the time) passing on the gridiron (from the Fox News story):
Baugh's reputation blossomed as a star high school athlete in football, baseball and basketball in Sweetwater. It began to grow during his college days at TCU.
It was there that he picked up the nickname "Slingin' Sammy" — but it wasn't for his passing. It was for the rockets he fired to first base as a shortstop and third baseman.
May his soul, and all the souls of the faithful departed, rest in peace.